Hyersulamrassias stability of generalized cauchy functional equation asghar rahimi abstract. Using the fixed point method, we prove the hyersulam stability of a cauchyjensen type additive setvalued functional equation, a jensen type additivequadratic setvalued functional equation, a generalized quadratic setvalued functional equation and a jensen type cubic setvalued functional equation. A new stability concept is introduced which generalizes the notion of the hyersulam stability. Research open access local stability of the pexiderized. The cauchy functional equation is shown to be stable in this more general sense.
The proof of this result is based on a new sandwich theorem proved also in this paper. In this paper, we study the stability of functional equations that has its origins with s. Beyer in 2 initiated the stability of functional equations in nonarchimedean spaces. Pexiderized cauchy functional equation, generalized hyersulam stability, jensen functional equation, nonarchimedean space 1. Solvability and stability of the cauchy equation relative to subsets of multidimensional euclidean spaces and tori. Stability of quadratic functional equations in generalized. Stability of a more general cubic functional equation in. Kim and rassias proved the stability of the eulerlagrange quadratic mappings.
We prove the hyersulam stability for this quartic functional equation by the directed method and the fixed point method on real banach spaces. Cauchys functional equation and a nonempty open set b. Random stability of quadratic functional equations. Generally a functional equation is said to be stable provided, for any function f satisfying the perturbed functional equation, there. Mohiuddine1 1 department of mathematics, faculty of science, king abdulaziz university, p. In fact they established stability of cauchy functional equations over padic fields. Additive selections and the stability of the cauchy functional equation.
Research article on the generalized hyersulam stability. On the stability of the squares of some functional equations. Hyers gave a first affirmative partial answer to the question of ulam for banach spaces. Stability problem of ulam for generalized forms of cauchy. In this paper, we improve their results and obtain better results for a cauchyjensen functional equation. At the other extreme, in the forefront of research, during the last two to three decades, the celebrated youngbaxter functional equation has been at the heart of many di. In this paper, we consider the following generalized quadratic functional equation with nindependent variables in the spaces of generalized functions. Pdf we give some stability results for the functional equation find, read and cite all the research you need on researchgate. Also, we establish new theorems for the generalized hyersulam stability of a cauchyjensen functional equation. In 2006, park and bae 9 obtained the generalized hyersulam stability of the cauchyjensen functional equation. The stability problem of functional equations originated from. Such equations are the subject of the book by kuczma 5. Pdf on the stability of the additive cauchy functional.
This article has been withdrawn consistent with elsevier policy on article withdrawal. On the stability of setvalued functional equations with. R adu, on the stability of the cauchy functional equation. Stability of the cauchy additive functional equation on. The aim of this paper is to generalize the above proposition by proving a stability result for the general functional equation. Fixed points and stability of the cauchy functional. Cauchyrassias stability of homomorphisms associated to a. Skof solved the hyersulam stability problem of the quadratic functional equation in banach spaces.
Pdf additive selections and the stability of the cauchy functional. We consider the stability, the superstability and the inverse stability of the functional equations with squares of cauchys, of jensens and of isometry equations and the stability in ulamhyers sense of the alternation of functional equations and of the equation of isometry. Stability of generalized jensen equation has been studied at. I advertised it not only to staff and phd students, but also to our 4th and 5thyear undergraduates, telling them that they should be wellplaced to understand it if they had a good grasp of our compulsory 3rdyear course honours analysis.
Using the fixed point method, we investigate the stability of the following generalization of cauchy s and the quadratic functional equations formula presented. Get a printable copy pdf file of the complete article 252k, or click on a page image below to browse page by page. Weight estimates for solutions of linear singular differential equations of the first order and the everittgiertz problem chernyavskaya, n. Cauchys equation on restricted domains the multiplicative cauchy equation. Semrl, the functional equation of multiplicative derivation is superstable on standard. Rassias for linear mappings by considering an unbounded cauchy difference. On the stability of functional equations in banach spaces core. Akkouchi, stability of certain functional equations via a fixed point of ciric.
A generalization of the rasssias theorem was obtained by gavruta 9 by replacing the unbounded cauchy difference by a general control function in the spirit of rassias approach. Stability of an alternative jensens functional equation. Using the method introduced in 3,jung 6 obtained a result for jensen. Pdf on stability of the cauchy functional equation in groupoids. On the stability of cauchyjensen type functional equation. On the stability of functional equations, aequationes math. Stability of the cauchy functional equation over request pdf. Aoki,onthe stability ofthe linear transformationin banach spaces, j. As an application, quasiadditive functions in the sense of jozef tabor are considered and stability and characterization theorems for them are obtained. On a cauchyjensen functional equation and its stability. We assumed that ma 0, where mx denotes the measure of a set.
Making use of the fundamental solution of the heat equation, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and fourier hyperfunctions. On the stability of the additive cauchy functional equation in random normed spaces. Radu, on the stability of the cauchy functional equation. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely. Introduction in 1940, ulam 1 gave a widerange talk before the mathematics club of the university of wisconsin in which he discussed a number of. Generalized ulamhyersrassias stability of a cauchy. Bae investigated the hyersulam stability of a cauchyjensen functional equation. On the normal stability of functional equations in. Analytically, the stability problem of functional equations originated from a question of ulam concerning the stability of group homomorphisms. Pdf stability of the cauchyjensen functional equation.
On supersingular primes of the elkies elliptic curve murabayashi, naoki, functiones et approximatio commentarii mathematici, 2019. Hyers, on the stability of the linear functional equation, proc. On the ulam stability of cauchy functional equation in ifnspaces a. Generalized ulamhyersrassias stability of a cauchy type. In this paper, using the direct method, we establish the hyersulamrassias stability of the generalized jensen functional equation of pexider type and the conditional stability on some. A fixed point approach to the stability of a cauchyjensen. Remarks on the cauchy functional equation and variations of it. On the stability of the linear functional equation in a. Stability of a generalization of cauchys and the quadratic functional equations muaadh almahalebi 1 journal of fixed point theory and applications volume 20.
The functional equation is called the cauchy additive functional equation. On the ulam stability of cauchy functional equation in ifn. Integrated cauchy functional equation with an error term and the exponential law by huamin gu south china normal university and university of pittsburgh and kasing lau university of pittsburgh summary. Equation 7, whose stability we have just proved, is often referred to as a functional equation in a single variable. We characterize the positive solutions of the functional equation fxlsx. Pdf a general fixed point method for the stability of cauchy functional equation. Functional equations in mathematical analysis springerlink. Asymptotic stability of the cauchy and jensen functional. Stability of the cauchyjensen functional equation in c.
In particular, every solution of the cauchy additive functional equation is. Rassias 37 proved the hyers ulam rassias stability of cauchys functional. Cauchys functional equation is the functional equation of linear independence. Preliminaries we recall some useful notions and results. Hyers theorem was generalized by aoki for additive mappings and by th. The authors3 obtained the stability of the cauchyjensen functional equation in the spirit of th. Box 80203, jeddah 21589, saudi arabia 2 department of mathematics, aligarh muslim university, aligarh 202002, india. Czerwik, functional equations and inequalities in several variables, world scientific publishing company, new jersey, hong kong, singapore and london, 2002. In 3, arriola and beyer initiated the stability of cauchys functional equation over padic fields. Stability of the cauchy jensen functional equation in algebras. On the stability of functional equations and a problem of ulam. Semrl, p the functional equation of multiplicative derivation is superstable on. Aoki 3 and rassias 4, 5 extended the hyersulam stability by considering variables for cauchy equation. Park considered the stability of quadratic mappings on banach modules.
Random pnormed space, quadratic functional equation, generalized hyersulam stability, direct method, fixed point method abstract in this paper, we investigate the generalized hyersulam stability on random normed spaces associated with the following generalized quadratic functional equation,where is a fixed positive integer via two methods. Pdf the stability of pexiderized cauchy functional equation. Stability of a cauchyjensen functional equation in quasi. R with positive measure, and fa \b any open set contains an open interval, so without loss of generality, we can assume that bis an open interval. Generalized hyersulam stability, cauchy functional equation, quadratic functional equation, nonarchimedean space, padic field. Jang, cauchyrassias stability of sesquilinear nquadratic mappings in banach modules, rocky mountain j. Jung, on the hyersulam stability of the functional equations that have the quadratic property, j. The stability problem of functional equations originated from a question of ulam concerning the stability of group homomorphisms.
Stability of the cauchyjensen functional equation in algebras. Pdf generalized stability of the cauchy functional equation. Stability of the exponential functional equation in riesz. Rassias 58 extended hyerss theorem by permitting the cauchy difference.
On the stability of a functional equation deriving from. On the stability of the additive cauchy functional. Czerwik,functional equations and inequalities in several variables, world scienti. Generalized stability of the cauchy functional equation. Full text full text is available as a scanned copy of the original print version. We obtain the generalized hyersulam stability of the cauchyjensen functional equation. Tabor asked about the stability problem for the following general linear functional equation. Najati, cauchyrassias stability of homomorphisms associated to a pexiderized cauchyjensen type functional equation, j.