Quadratic Functional Equation Research Articles

Fuzzy normed spaces and stability of a generalized quadratic functional equation

In this paper, we acquire the general solution of the generalized quadratic functional equation $ \begin{aligned} \sum\limits_{1 \leq a \lt b \lt c \leq m}\varphi\left(r_{a}+r_{b}+r_{c}\right)& = (m-2)\sum\limits_{1\leq a \lt b\leq m}\varphi\left(r_{a}+r_{b}\right) &\quad-\left(\frac{m^{2}-3m+2}{2}\right)\nonumber \sum\limits_{a = 1}^{m}\frac{\varphi\left(r_{a}\right)+\varphi\left(-r_{a}\right)}{2} \end{aligned} $ where $m\geqslant 3$ is an integer. We also investigate Hyers-Ulam stability results by means of using alternative fixed point theorem for this generalized quadratic functional equation.

Ulam stability for finite variable quadratic functional equation in Banach algebra

Here, we define a cube sum labeling and cube sum graph. Let G be a (p, q) graph. G is said to be a cube sum graph if there exist a bijection f : V (G) → {0, 1, . . . , p − 1} such that the induced function f ∗ : E(G) → N given by are all distinct. In this paper, we developed the concept of cube sum labeling of some family of graphs like paths, cycle, stars, wheel graph, fan graphs are discussed in this paper.

Stability of 4-variable quadratic functional equation in Banach spaces

In this work, authors investigate the generalized Hyers-Ulam stability of the 4-variable quadratic functional equation of the form

Stability results of the functional equation deriving from quadratic function in random normed spaces

<abstract> <p>The authors introduce a finite variable quadratic functional equation and derive its solution. Also, authors examine its Hyers-Ulam stability in Random Normed space(RN-space) by means of two different methods.</p> </abstract>

Solution to Kim‐Rassias's question on stability of generalized Euler‐Lagrange quadratic functional equations in quasi‐Banach spaces

By using Aoki‐Rolewicz Theorem on p‐normalizing a quasi‐normed space, we prove stability results for Euler‐Lagrange quadratic functional equations in quasi‐Banach spaces. These results improve stability results and give the answer to Kim‐Rassias's question.

Measure zero stability problem of a generalized quadratic functional equation

Let X be a normed space, Y be a Banach space and $$f,g: X\rightarrow Y$$. In this paper, we investigate the Hyers–Ulam stability theorem for the generalized quadratic functional equation $$\begin{aligned} f(kx+y)+f(kx-y)=2k^2g(x)+2f(y) \end{aligned}$$in a set $$\Omega \subset X\times X$$, where k is a positive integer. By the Baire category theorem, we derive some consequences of our main result.

A New Fixed Point Theorem and a New Generalized Hyers-Ulam-Rassias Stability in Incomplete Normed Spaces

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results.

Approximate solutions of a quadratic functional equation in 2-Banach spaces using fixed point theorem

Approximate solutions of a quadratic functional equation in 2-Banach spaces using fixed point theorem

Random Stability of Quadratic Functional Equations

In this paper, we investigate the generalized Hyers-Ulam stability on random -normed spaces associated with the following generalized quadratic functional equation ,where is a fixed positive integer via two methods

Some generalized quadratic functional equations on monoids

We solve two Pexiderized extensions of the quadratic functional equation on monoids. The equation is extended by replacing the group inversion in the classical quadratic equation with an involution, which may be either homomorphic or anti-homomorphic. Also two extensions of the Drygas functional equation are solved in the same setting. In all results it is assumed that at least one unknown function is central.

On asymptotic stable solutions of a quadratic Erdélyi-Kober fractional functional integral equation with linear modification of the arguments

Abstract We consider the solvability and asymptotic stability of a quadratic Erdelyi-Kober fractional functional integral equation with linear modification of the argument. In the Banach space of functions which are bounded and continuous on R + : = [ 0 , ∞ ) , Schauder’s fixed point theorem and the measure of noncompactness in this space are the main tools used to prove our main result.

On stability and hyperstability of additive equations on a commutative semigroup

We prove the stability and hyperstability results for the additive functional equations on restricted domains (subsets of a commutative semigroup). The proof is based on the modified Brzdek's fixed point theorem. Next, we introduce the concepts of modulus-additive and approximately modulus-additive functions and deduce, from our main theorem, some stability and hyperstability outcomes for the modulus-additive equation. We also obtain some results for the radical quadratic functional equation. Finally, we point out some faulty assumptions in a result of Aiemsomboon and Sintunavarat [1].

An Orthogonal Stabilization of Quadratic and Generalized Quadratic Functional Equations

An Orthogonal Stabilization of Quadratic and Generalized Quadratic Functional Equations

Stability of two variable pexiderized quadratic functional equation in intuitionistic fuzzy Banach spaces

The present work is about the stability of a Pexiderised quadratic functional equation. The study is in the framework of intuitionistic fuzzy Banach spaces. The approach is through a fixed point method. The stability studied is Hyers-Ulam-Rassias stability type.

Almost additive-quadratic-cubic mappings in modular spaces

We introduce and obtain the general solution of a class of generalized mixed additive, quadratic and cubic functional equations. We investigate the stability of such modified functional equations in the modular space Xρ by applying the ∆2-condition and the Fatou property (in some results) on the modular function ρ. Furthermore, a counterexample for the even case (quadratic mapping) is presented.

On the Complete 2-Normed Stabilization of Cubic and Quadratic Functional Equations

On the Complete 2-Normed Stabilization of Cubic and Quadratic Functional Equations

Classification of generalized ternary quadratic quasigroup functional equations of the length three

A functional equation is called: generalized if all functional variables are pairwise different; ternary if all its functional variables are ternary; quadratic if each individual variable has exactly two appearances; quasigroup if its solutions are studied only on invertible functions. A length of a functional equation is the number of all its functional variables. A complete classification up to parastrophically primary equivalence of generalized quadratic quasigroup functional equations of the length three is given. Solution sets of a full family of representatives of the equivalence are found.

Nice results about quadratic type functional equations on semigroups

Let $$(S,+)$$ be an abelian semigroup, let $$\sigma $$ be an involution of S, let X be a linear space over the field $${\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}$$ and let $$\mu $$,$$\nu $$ be linear combinations of Dirac measures. In the present paper, we find the general solution $$f:S\rightarrow X$$ of the following functional equation $$\begin{aligned} \int _{S}f(x+y+t)d\mu (t)+\int _{S}f(x+\sigma (y)+t)d\nu (t)=f(x)+f(y), \ \ \ x,y \in S, \end{aligned}$$in terms of additive and bi-additive maps. Many consequences of this result are presented.

Alienation of the Quadratic and Additive Functional Equations

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation (*) $$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$ resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.

Approximate solution of generalized inhomogeneous radical quadratic functional equations in 2-Banach spaces

In this paper, using Brzdȩk and Ciepliński’s fixed point theorems in a 2-Banach space, we investigate approximate solution for the generalized inhomogeneous radical quadratic functional equation of the form \t\t\tf(ax2+by2)=af(x)+bf(y)+D(x,y),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$f \\bigl(\\sqrt{ax^{2}+by^{2}} \\bigr)=af(x)+bf(y) + D(x,y), $$\\end{document} where f is a mapping on the set of real numbers, a, binmathbf {R}_{+} and D(x,y) is a given function. Some stability and hyperstability properties are presented.